Closure & Closed Sets in metric space

[kingwinner]

Definition: Let F be a subset of a metric space X. F is called closed if whenever is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limits of sequences in F) The closure of F is the set of all limits of sequences in F.

Claim 1: F is contained in the clousre of F.
Claim 2: The closure of F is closed.

How can we prove these formally?

For claim 1, I think we have to show x E F => x E cl(F).
For claim 2, we need to prove that cl(F) contains all limits of sequences in cl(F).
But how can we prove these?

I know these are supposed to be basic facts, but my book never gives examples of how to prove these from the definitions given above...and I have no clue...
Any help is appreciated!

 

[VeeEight]

For claim 1, yes, you want to show that if x is in F, then x is in the closure of F. For some element x in F, can you find a sequence in F that converges to x?

For claim 2, simply apply the definition of closure to what is being asked

 

[kingwinner]

1) So we have to show that if x E F, then x is the limit of a sequence in F??
How can we show it? I can't figure it out...

2) The closure of F is the set of all limits of sequences in F.
But we need to prove that cl(F)[/color] contains all limits of sequences in cl(F)[/color]. How?

Can somebody explain a little more, please?
Thanks!

 

[VeeEight]

(1) Think about constant sequences.

(2) Clearly cl(F) = F u F' where F' is the set of limit points of F. Use this and think about how 'adding' F' to F changes the sequences in F - all you are doing is 'completing' the sequences in F, you are not introducing anything new, so there would be no more potential limit points to consider.

 

[Tinyboss]

(2) Let {c_n} be a convergent sequence in cl(F). We want to show that it has its limit in cl(F). We can find a sequence {a_n} of elements in F, such that for every n, d(a_n,c_n)<1/n (because c_n is the limit of some sequence in F). Therefore lim c_n = lim a_n in cl(F).